Questions Further Paper 2

Use the method of differences to show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { ( r - 1 ) r ( r + 1 ) } = \frac { 1 } { 4 } - \frac { 1 } { 2 n } + \frac { 1 } { 2 ( n + 1 ) }$$ [5 marks]
13
Find the smallest integer $n$ such that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { ( r - 1 ) r ( r + 1 ) } > 0.24999$$

Given that $\mathbf { M }$ is a non-singular matrix, find $\mathbf { M } ^ { - 1 }$ in terms of $k$
14
State any restrictions on the value of $k$ 14
Using your answer to part (a), show that the solution to the set of simultaneous equations below is independent of the value of $k$ $$\begin{array} { r l c c } 5 x + 2 y + c & = & 1
6 x + 3 y + ( 2 k + 3 ) z & = & 4 k + 3
2 x + y + 5 z & = & 9 \end{array}$$

On the diagram above, sketch the graph of $y = \left| x ^ { 2 } - 4 x \right|$, including all parts of the graph where it intersects the line $y = 5 - x$
(You do not need to show the coordinates of the points of intersection.) 15
Find the solution of the inequality $$\left| x ^ { 2 } - 4 x \right| > 5 - x$$ Give your answer in an exact form.
[0pt] [4 marks]

Write down the value of $a$ and the value of $b$ 16
The diagram shows the graph of $y = \mathrm { f } ( x )$ and its asymptotes.
The shaded region $R$ is enclosed by the graph of $y = \mathrm { f } ( x )$, the $x$-axis and the $y$-axis.
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1. The shaded region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. 16
(ii) The shaded region $R$ is rotated through $360 ^ { \circ }$ about the $\boldsymbol { y }$-axis to form a solid.
Find the volume of this solid.
Give your answer to three significant figures.
[0pt] [4 marks]

Write down the equation of the locus of $C$ in the form $$| z - w | = a$$ where $w$ is a complex number whose real and imaginary parts are integers, and $a$ is an integer.
17
It is given that $z _ { 1 }$ is a complex number representing a point on $C$. Of all the complex numbers which represent points on $C , z _ { 1 }$ has the least argument. 17
1. Find $\left| z _ { 1 } \right|$
  Give your answer in an exact form.
  17
(ii) Show that $\arg z _ { 1 } = \arcsin \left( \frac { 6 \sqrt { 3 } - 2 } { 13 } \right)$

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Show that $$I _ { n } = \left( \frac { n - 1 } { n } \right) I _ { n - 2 } + \frac { 1 } { n \left( 2 ^ { \frac { n } { 2 } } \right) } \quad ( n \geq 2 )$$ 20
Use the result from part (a) to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { 6 } x d x = \frac { a \pi + b } { 192 }$$ where $a$ and $b$ are integers to be found.
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Questions Further Paper 2 (287 questions)